让开吧，氛围编码，数学领域迎来氛围证明

嗨。

Hey.

我是弗洛拉·利希特曼，欢迎收听《科学星期五》。

I'm Flora Lichtman, and you're listening to Science Friday.

几年前ChatGPT刚推出时，最让人惊叹的功能之一就是它的写作能力。

When ChatGPT came out a few years ago, one of its most impressive features was its writing abilities.

但你可能还记得，除了生成七根手指的手图片外，它常被嘲笑的是连基本的算术都做不好。

But something it was often ridiculed for, you may remember, other than making images of seven vancored hands, was its inability to do basic math.

二加二并不总是等于四。

Two plus two did not always equal four.

但最近，情况已经不一样了。

But recently, things have changed.

去年，谷歌和OpenAI的模型在国际数学奥林匹克竞赛中摘得金牌，超过了世界上一些顶尖的高中生数学选手。

Last year, Google and OpenAI's models bagged gold medals at the International Mathematical Olympiad, edging out some of the best high school mathletes in the world.

现在有些专家认为，AI的进步已经快到可能对整个数学领域构成根本性威胁了。

And now some experts say AI is getting so good, it could pose an existential threat to the field.

今天请来两位深入思考过这个问题的数学家，约翰斯·霍普金斯大学的艾米莉·里尔博士和多伦多大学的丹尼尔·利特博士。

Here with a perspective are two mathematicians who've thought a lot about this, doctor Emily Riehl from Johns Hopkins University and doctor Daniel Litt from the University of Toronto.

艾米莉，丹尼尔，欢迎来到《科学星期五》。

Emily, Daniel, welcome to Science Friday.

你好。

Hey.

谢谢。

Thank you.

好的。

Alright.

至少在我关注的那些科技圈媒体里，最近总看到一些很夸张的说法，说AI突然彻底改变了数学。

We have seen, at least in the the nerdy media that I consume, this these very big claims that AI suddenly is revolutionizing math.

它正在重新定义‘数学家’这个身份。

It's fundamentally changing what it means to be a mathematician.

我想听听你们的看法。

I want your takes on that.

艾米丽，先从你开始吧。

Emily, let's start with you.

我觉得，从职业数学家的角度来看，我们得先评估一下人工智能在数学家职业生涯中处于什么阶段。

I think from the perspective of a professional mathematician, we're trying to evaluate where AI is on the trajectory of a mathematician's life.

所以，我们最早在数学里接触到它，是在学校里做那些有标准答案的题目，可能涉及几何图形，也可能是一些算术或代数题。

So we're first encountered in mathematics at school where you're solving problems that have a numerical answer, maybe involving some geometric figures, maybe involving some arithmetic or some algebra.

正如你刚才提到的，人工智能以前在这些题目上表现得很差。

As you mentioned already, AIs used to be very bad at those problems.

现在它们的表现好多了。

And they're much better now.

最近，AI在处理更高级的竞赛题方面表现不错，比如针对数学专业本科生的证明题竞赛。

More recently, AIs have been doing well with more advanced contest level problems, contests for undergraduate mathematics majors where they're writing proofs.

过去这一年让人既兴奋又有点担心的是，它们开始接受更具研究水平的难题测试，也就是那些职业数学家真正关心的问题。

And what's exciting and perhaps scary in this past year is they're starting to be tested on more research level problems, so problems that are of interest to people who get paid to do mathematics.

丹尼尔，你怎么看？

Daniel, what do you think?

是的。

Yeah.

所以，说AI工具正在彻底改变数学，这一点目前还远未实现。

So so the specific claim that AI tools are revolutionizing mathematics definitely has not come true yet.

但我认为已经出现了一些明显的迹象，表明它将来会带来重大影响。

I think there are leading indicators that it will be very significant.

它确实正在改变包括我在内的专业数学家的工作方式。

It's definitely changing the way professional mathematicians, including me, work.

不过，Emily提到的那些AI被用于研究级数学问题的案例，确实有几例成功的事例。

But the the cases Emily mentioned of AI is being tested on on research level mathematics, there are a few success stories.

我觉得这些是令人振奋的早期信号。

I think those are exciting leading indicators.

但目前为止，还没有任何例子能称得上是革命性的。

There's not any example of of anything that I would consider to be revolutionary yet.

未来一两年会怎样，谁也说不准。

Who knows what the next year or two will bring?

所以我对这一点感到非常兴奋。

So I I'm excited about that.

我认为这个故事中很重要的一点是，数学不仅仅是解决问题。

And I think an important part of the story is mathematics is not just about solving problems.

即使是研究型数学，也不只是关于解决问题。

Even research mathematics is not just about solving problems.

解决问题当然可以很美妙。

Mean, solving problems can be beautiful.

但提出问题、发现哪些定理值得去证明，同样重要，这才是推动整个领域前进的核心。

But actually stating the problems, figuring out what is an interesting theorem to try and prove is equally important, really what drives the field forward.

到目前为止，我们还没看到AI能提出新的数学问题，或者开辟值得探索的全新数学领域。

And we haven't seen any examples there where AIs are coming up with problem statements or new mathematical universes to explore.

就像挑选出我们该去努力解决的问题那样。

Like picking the things that we should be trying to solve.

对。

Yes.

我的意思是，AI有没有解决过人类一直没解决的问题？

I mean, is AI solving problems that people haven't been able to solve?

这种情况发生过吗？

Has that happened?

你觉得这种事快实现了吗？

Does it seem like it's on the horizon?

要回答这个问题，一个角度是，数学领域还有太多等待发现的东西，全球有无数数学家在合作或独立地努力做出新发现、证明新定理。

I one way to answer that is that, there is so much math that still needs to be discovered, and there are a lot of mathematicians working together and individually all around the world trying to make new discoveries and prove new theorems.

但我们的人数远远不够，不可能解决所有问题。

But there aren't enough of us to solve every problem.

特别是，网上有一个著名的数学问题数据库，是数学家保罗·埃尔德什收集整理的。

And so in particular, there's a famous database of problems tied to the mathematician Paul Erdosch that has been collected online.

其中一些问题非常有名，备受关注，但还有很多，我们甚至不确定有没有人认真研究过。

And some of them are quite famous and have received a lot of attention, but others, we don't actually know if anybody's given a serious effort to them.

我知道，像艾米莉提到的，大概有三个问题是由AI完全独立解决的，另外六七个问题是在没有文献先例的情况下，由人类借助AI工具解决的，还有一些问题是AI从现有文献中提取出了解法。

So I'm aware of, I think, maybe three problems, like Emily mentioned, that have been solved fully autonomously and maybe another six or seven without prior solutions in the literature that were were solved by a a human, with AI help, and then a number where sort of solutions were kind of extracted from the literature by AI tools.

我觉得这真是个令人振奋的进展。

I think that's a really exciting development.

不过话说回来，我们应该把这看作一个积极的信号，说明很快这些工具就能帮我们解决更多人们长期努力却未果的难题。

I think that said, like, we should view this as, like, an exciting sign that, like, soon the tools will be useful to help us solve maybe more interesting questions that people have devoted effort into trying to answer.

你提到‘令人振奋’这个词，其实挺有意思的。

You know, it's interesting that you use the word exciting.

作为对比，去年我们请过数学家史蒂夫·斯特罗加茨做客，他说过一句话，我想听听你的看法。

Just as a contrast, we had mathematician Steve Strogatz on last year, and he said something I want, your thoughts on.

我觉得我们真正理解数学的日子可能不多了，用不了多久，计算机就能生成出我们完全看不懂但又无比正确的数学成果。

I think the days when we will understand math may be numbered, that that it will not be far in the future when computers are producing really impressive math that we will not understand.

那些成果是对的，但就像神谕一样，只告诉我们答案，我们却完全不明白为什么。

And it will be correct, but it'll be like they're oracles just telling us the truth and we can't understand it.

我们只能坐在那儿，张着嘴发呆。

We'll just just be sitting there with our mouth open.

艾米莉？

Emily?

我完全同意斯蒂夫的观点，如果我们有个全知的神谕来告诉我们哪些定理为真、哪些为假，数学家们一定会觉得这很令人失望。

I agree a 100% with Steve that if we had an oracle to tell us which theorems are true and which theorems are false, that would be unsatisfactory to mathematicians.

我们确实喜欢看到证明过程揭示了新的东西——原来某个命题是对的，或者错了。

We do like it when a proof tells us something new, that something is true or something is false.

但如果这个证明没能解释出背后的深层道理，来支撑这个新结论，数学家们就会去寻找另一个能说清楚的证明。

But if the proof doesn't also explain something deep to sort of justify the new conclusion, then mathematicians will go and search for a different proof that does.

人们甚至会因为重新证明了老定理而获得赞誉，其程度不亚于证明全新定理。

People get celebrated for reproving old theorems almost as much as for proving new theorems.

我设想，如果AI发现了一个让人类难以理解的定理证明，那么人类很可能会尝试自己去证一遍，说不定就在这个过程中找到了新的证明方法。

What I would imagine if an AI discovers a proof of a theorem that humans have difficulty understanding is that humans will then try to prove it themselves and may discover a new proof that way.

但我也想提一下，AI生成证明其实有两种不同的方式。

But I also want to bring into the conversation the fact that there are sort of two different modes in which an AI can produce a proof.

如果这个证明完全超出了人类的理解能力，而且只是以自然语言的形式输出，比如像ChatGPT对话那样，那就有理由认为，数学家们不如直接把它扔进废纸篓，转而要求AI给出一个形式化的证明。

And if it's really a proof that is beyond the level of human understanding and it's communicated just in natural language, so as the output of a ChatGPT session, there's an argument that actually mathematicians should just throw it in the garbage and ask instead for an AI to produce a proof that is formalized.

形式化，就像数学里那种严格的符号语言。

Formalized like in math, in the language of math.

真的吗？

Is that right?

没错。

Right.

我觉得数学领域有个特别令人兴奋的点，其他领域很少有类似的情况，那就是我们有几种被广泛信赖的软件工具，叫作计算机辅助证明系统，它们是由人类专家专门设计的，完全不是用AI做的，能够逐行检查一条精确写出来的数学证明的逻辑是否正确。

So a thing that's exciting about the possibility of generating AI for mathematics that I think doesn't exist in quite the same way in other fields is that there are these trusted software programs called computer proof assistants that are engineered by human experts, so not produced with AI in any way, that will take a very precisely written mathematical proof and check the logic line by line.

很多数学家都担心，我们会迎来大量AI生成的垃圾内容，一堆新预印本声称自己证明了那些著名定理。

So a concern that many mathematicians share is that we're going to get a lot of AI slop, a lot of new preprints purporting proofs to very famous theorems.

但这些证明又长又复杂，需要极高的专业水平才能评估。

But they're long and complicated and require a ton of expertise to evaluate.

人类根本不可能一一核实这些证明的准确性。

And humans will just never be able to judge these proofs for accuracy.

不过就像大型语言模型能生成Python代码一样，我们也可以让大型语言模型把证明写成这些计算机辅助系统所使用的语言，而不是用英语。

But just as a large language model can produce Python code, a large language model can be asked to write the proof not in English, but in the language of these computer proof assistants.

这样一来，至少审稿过程的一部分就能自动化了，可以交给计算机来处理。

And then at least some aspect of the refereeing can be automated, can be outsourced to a computer.

我觉得，比起只用自然语言，和AI在这一领域互动更有意义。

And I think that's a more productive realm to interact with AI than just the natural language realm.

丹尼尔，我想听听你对这一点的看法，还有AI会不会成为数学领域的神谕这个想法。

Daniel, I wanna hear your thoughts on on that and also on this idea that AI will become oracles for us in math.

是的。

Yeah.

我确实认为，从长远来看，AI工具在证明数学命题方面可能会超越人类。

So so I definitely think it's possible that in the long term, AI tools will become better at proving mathematical statements than human beings.

我们还没达到那一步，但未来是有可能的。

We're not there yet, but it could happen.

就像艾米莉说的，这和我作为数学家的目标——理解数学——有点不太一致。

As Emily suggests, that would be a little bit unsatisfying with my goal as a mathematician is to understand mathematics.

不过，我也说不准，AI生成了精彩而复杂的数学证明，就一定意味着我们无法理解它们。

That said, I don't think it's not clear to me that that AI tools producing you know, amazing and long proofs of of mathematical statements would mean that we couldn't understand them.

既然AI能证明出这么了不起的定理，为什么就不能把它讲得清清楚楚呢？

Like, why wouldn't we expect an AI tool able to prove an awesome theorem also to be able to explain it to us really well?

所以，说实话，只要我有足够的动力去理解计算机生成的数学内容，我就相信自己一定能搞懂。

So, yeah, I I'm kind of I feel confident that if I'm sufficiently motivated to understand a piece of mathematics produced by a computer, that I'll be able to do it.

这正是我的目标。

And that's kind of my goal.

但我认为，AI生成的数学和人类生成的数学之间，存在一个很大的道德问题。

But I guess I mean, there's a there's a big problem, a kind of a moral problem with AI generated mathematics and human generated mathematics.

在数学界，有一个非常强的共识：除非你确信某个证明是正确的，否则你不能把它发布到网上，也就是我们那个预印本服务器上。

So there's a very strong norm in the mathematics community that you do not post a proof to the internet, to the archive, which is our mathematics preprint server, unless you believe that it's true.

但我们不能对大型语言模型提出同样的要求，因为它们完全不可靠。

And we can't ask the same things of large language models because I mean, they're completely untrustworthy.

它们没有信念。

They don't have beliefs.

我对大型语言模型并没有真正的心理理论。

I don't really have a theory of mind of large language models.

所以这类比喻其实都不太准确。

So these sort of metaphors are imperfect.

所以我认为，对于AI写的证明，我们应该提出比人类写的更高的要求，因为我们缺乏那种信任和信念的准则。

So I think we should demand more of a proof written by an AI than a proof written by a human because we don't have this sort of norm of trust, this norm of belief.

AI给我们提供了一个很长的证明，就像丹尼尔说的，AI的工作还没做完。

So AI gives us a large proof that is difficult to understand, as Daniel said, the AI has not finished its job.

它还应该给我们一个解释。

It should also give us an explanation.

但也许这个解释也得用这种形式化的方式呈现，这样我们才能借助计算机工具启动整个过程，虽然这会非常耗时。

But maybe that explanation should also be in this formal realm so we can use computer tools to start the process, which will be very long.

我们得休息一下。

We have to take a break.

但回来之后，我想聊聊‘感觉证明’。

But when we come back, I wanna talk about vibe proving.

你听过‘感觉编程’吧。

You've heard of vibe coding.

那你听过‘感觉证明’吗？

Have you heard of vibe proving?

你会的。

You will.

别走开。

Stay with us.

喂。

Hey.

我是弗洛拉。

It's Flora.

我知道你可能听过很多次了，但这可不是套话。

I know that you have heard this before, but it's not a line.

科学星期五能继续做下去，全靠你们这些听众的支持。

Science Friday really can only continue with support from you, our listeners.

我特别喜欢制作和讲述这些故事。

I love making and telling these stories.

如果你喜欢听，欢迎去 sciencefriday.com/donate 捐助我们。

And if you like listening to them, please go to sciencefriday.com/donate to make a donation.

加入我们吧。

Join us.

支持科学星期五和公共媒体的价值，帮助我们继续激发好奇心，传播科学的乐趣。

Stand up for the value of science Friday and public media, and help us continue to spark curiosity and spread the joy of science.

捐款快速、简单又安全。

Donations are fast, easy, and secure.

请前往 sciencefriday.com/donate 捐款，谢谢您。

Just go to sciencefriday.com/donate, and thank you.

几十年来，我一直听人说市场能解决气候变化问题。

I've been hearing for decades that the markets can solve climate change.

如今，我们拥有的市场解决方案激励比以往任何时候都多，但排放量却还在上升。

Today, we have more incentives for market solutions than ever, and emissions are rising.

在《钻探：碳牛仔》这一季中，我们将讲述三种市场方案在一个跨国闹剧中相互碰撞的故事。

On this season of Drilled Carbon Cowboys, the story of three market solutions colliding in one multinational moondoggle.

得让布鲁斯去确认一下那人的付款信息。

Gotta get Bruce with the guy's credit.

他们是共和党人。

They're Republicans.

他们根本不在乎这些事。

They don't give a shit about any of this stuff.

你在哪里听播客，都能听到我们。

Listen anywhere you get podcasts.

好的。

Okay.

我们听说过‘氛围编程’。

So we've heard of vibe coding.

数学里有类似的东西吗？

Is there an equivalent in math?

比如，我能这么问吗？

Like, could I ask it?

我能说：我不太懂该怎么弄吗？

Can I be like, I don't really understand how to do this?

你能解决这个问题吗？

Can you can you solve this problem?

是的

Yeah.

你当然可以这么问。

You could certainly ask that.

有时候它会给出正确答案，有时候却会出错。

And then, well, sometimes it would give you the right answer, and sometimes it would give you a wrong answer.

但人还是得花不少功夫去核对答案对不对。

There's still a lot of work to do for a human, which is you have to check the answer is correct.

这挺难的。

That's hard.

你可以靠直觉去推导，但很多时候直觉会出错。

So you can vibe prove something, and then, well, often it will be vibe wrong.

不过有时候它确实能对，有时候还特别有用。

And but sometimes it'll be correct, and sometimes it'll be really useful.

所以，我觉得这已经开始成为数学研究流程的一部分了。

So, yeah, I think this is starting to be a part of the mathematical workflow.

归根结底，还是得有人对结果的正确性负责，这一点非常重要。

It's still important at the end of the day that a human is there taking responsibility for the correctness of the results.

我觉得我们已经能看到一些这方面出错的例子了。

And I think you we can see some examples of this going wrong.

比如去年九月，我统计了在预印本平台arXiv上，标题或摘要里含有‘霍奇猜想’字样的论文数量。

Like, in September, I I counted the number of papers put on the archive, which is the main math preprint server with the words Hodge conjecture in the title or abstract.

霍奇猜想是七大千禧年难题之一。

The Hodge conjecture is one of the millennium problems.

我算下来大概一共十二篇左右。

I think there were about 12 total.

据我判断，其中十一篇都是AI生成的胡言乱语。

And and as far as I could tell, 11 of them were were nonsense generated by generated by AI tools.

就像幻觉那样？

Like hallucinations?

我们说的就是这个吗？

Is that what we're talking about?

我觉得是有人想证明霍奇猜想，毕竟那上面挂着一百万美元的奖金。

Well, I think it was a human who wanted to prove the Hodgkin's conjecture because there's a million dollar prize associated to it.

于是他们对Claude说：你来证明一下霍奇猜想。

So they said, Claude, prove the Hodge conjecture.

别出错。

Don't make any mistakes.

然后就把生成的结果直接发到了预印本服务器上，结果根本不对。

And then they posted whatever came out of that to the archive, and it it wasn't correct.

对。

Right.

丹尼尔想说的是，这种靠感觉来证明，就跟靠感觉写代码一样有风险。

And what Daniel is alluding to is there's a danger with vibe proving, just like with vibe coding.

对专业程序员来说，靠感觉写代码或许能提升效率。

So vibe coding for an expert programmer can be an enhancement of their workflow.

但这也会让一些完全新手的人，比如我自己在编程方面，误以为自己能做出远超实际能力的成果。

But it would also allow somebody who's a total novice, so I'll put my hand up here in the programming context, to to think that they can achieve more than they actually can with computer programming.

但要分辨什么时候模型在装样子、什么时候它说的是对的，需要很多专业知识。

But it requires a lot of expertise to know when when it's bluffing and and when it's sound.

关于这一点，我想说的是，这些模型造假的方式，其实和人类撒谎、蒙混过关的方式非常相似。

One one thought on this is that, actually the ways the models bluff are remarkably similar to how humans bluff and approve.

真的吗？

Really?

你再详细说说。

Say more about that.

是的。

Yeah.

好吧。

So okay.

你可以想想，比如让一个本科生做作业题，他们常常会故意跳过一些自己不会的步骤，指望老师看不出来，有时候还真能蒙混过关。

You can ask well, first of all, you can ask an undergrad to prove something on their homework, and often they'll, you know, leave some things out that they don't know how to do and hope to get full marks, and sometimes that works out.

即使你直接问一位专业数学家当场证明某个命题，他也可能只有个大致想法，没把所有细节都推敲清楚，有时候对，有时候不对。

Even if you ask a professional mathematician just off the top of their head, prove something, they might have some kind of general idea and not think through all the details, and sometimes that'll be right and sometimes not.

所以我觉得，看AI做数学题特别有意思，它在某些方面简直跟人一模一样。

So I I think it's actually remarkable to watch the models doing math and see, in some ways, how human they are.

对。

Mhmm.

我的意思是，证明辅助工具能揪出AI数学里的漏洞吗？

I mean, can the proof assistant sort of root out the AI math slop?

能。

Yes.

我觉得可以。

I I think so.

不过这里也有一些限制。

So there are some caveats there.

首先，它要求精确得多。

Firstly, it's a lot more exacting.

所以在使用证明助手时，你不能敷衍了事，跳过几步推理。

So you can't sort of wave your hands and skip a few steps in a proof assistant.

它根本不会接受这种做法。

It will not accept it.

它会明确标注出哪些地方还没有得到充分论证。

It'll make a note of exactly what has not yet been justified.

所以我确实觉得它能帮我们筛掉不少粗糙的推导。

So I do think it can help weed out some slop.

尤其是一些AI初创公司，正努力提升编写形式化证明的能力，它们通过证明助手的反馈来评估不完整的证明。

And in particular, there are some AI startup companies that are aiming to get better at writing formal proofs, they are training by using the feedback from the proof assistant to evaluate partial proofs.

证明助手的设计本就是交互式的，会逐行告诉你每一步是对是错。

The proof assistant is meant to be used interactively and will give you some information about whether each line of your proof is correct or incorrect.

这些信息完全可以整合进AI的工作流程，帮助优化证明的搜索过程。

And so that information could be integrated into an AI workflow to improve the search for proofs, for instance.

这些工具会不会改变成为一名优秀数学家所需具备的技能？

Do these tools change what skills are required to be a great mathematician?

我觉得如今当数学家最困难的一点，就是这个领域的范围实在太广了。

I think one of the things that's most challenging about being a mathematician today is just the breadth of the field.

我和丹尼尔研究的方向差别太大了，恐怕连对方最近的论文都很难看懂。

So Daniel and I are in different enough research areas that I think we would have difficulty understanding each other's most recent papers.

我自己也觉得，就算只盯着我这个小领域，想跟上最新文献都很难——每个月都有太多令人兴奋的新成果出来，可我的阅读速度根本赶不上。

And I also know that it's very difficult to keep on top of the literature, even in my own subject area, because there's a lot of exciting work that gets finished every month, and I I don't read at the pace that I would like to.

所以我对这些工具挺乐观的，觉得它们能帮我更好地跟上领域里的进展。

So a thing that I'm optimistic about is that these tools will help me stay more on top of the literature.

是的。

Yeah.

我觉得这些工具对我学新东西特别有帮助。

I I found them really useful to learn things that I needed.

比如去年夏天，我通过ChatGPT-3学了超复几何，本来我可以找位专家问，但人家肯定早就不耐烦了，AI却一点不嫌烦。

Like, over the summer, I learned from ChatGPT o three what what about hypercalar geometry, which is something that, you know, I I could have gotten a human expert and pestered them, but they would have gotten annoyed with me much more rapidly than than the AI tool did.

我觉得吧，任何新技术出现，都会让一些过去必须掌握的技能变得不再必要，同时又打开新的可能性，数学领域的AI工具也一定会这样。

I think that, you know, anytime a new technology is developed, it obviates some skills that that by by automating them that that humans previously needed, and then it also opens up some new capabilities, and I I absolutely expect that to happen with AI tools for mathematics.

现在来说，要准确判断哪些技能会变得不那么重要，又会催生哪些新能力，还为时过早。

Right now, I think it's sort of too early to figure out exactly what skills will become less necessary and what new capabilities will open up.

但我相信，这会带来很多新的可能性。

But I'm hopeful that it will open up a lot of new possibilities.

我觉得，人们对‘优秀的数学家’是什么样子，其实有很多误解。

I mean, I think, you know, there's a lot of misconceptions about what a great mathematician looks like.

有些人觉得，这种天赋在很小的时候就能看出来。

There's some stereotypes that this is something that's identifiable when you're very young.

事实上，像我这样41岁的人，早就过了拿菲尔兹奖的年龄了。

And in fact, maybe when you're my age, I'm 41, which means I'm no longer eligible for the Fields Medal.

但现在的我，比二十年前厉害多了，因为这二十年我一直在认真思考数学。

But I'm a much better mathematician now than I was twenty years ago because I have spent so much time thinking about mathematics in the last twenty years.

这其中一部分是阅读文献，一部分是尝试解决问题却失败了，但这些失败让我积累了经验——以后遇到类似问题时，我能更清楚地判断哪些方向值得尝试，哪些大概率走不通。

And some of that is reading the literature, some of that is attempting to solve a problem and failing, but failing in a way that when a similar problem comes up later, I will have better intuition about what avenues might be productive and what might be not productive.

所以我觉得，真正造就一位优秀数学家的，是时间，是对数学的持续投入。

So I, you know, I really think what makes a great mathematician is time and dedication to mathematics.

只要有人真正爱上这门学科，并且有幸能全身心投入，就一定能做出一些很了不起的成就。

You know, anybody who falls in love with the subject and is fortunate enough to be able to devote themselves to it is gonna be able to achieve some pretty cool things.

是的。

Yeah.

所以，嗯，到底是什么造就了一位伟大的数学家呢？

So so, well, yeah, what makes a great mathematician?

当然，像艾米莉说的那样，对这门学科的热爱和内在动力非常重要。

So, certainly, I I think loving the subject motivation, like Emily was talking about, are very important.

你确实需要一定的技术能力，而人们通常一想到数学能力强，就会想到这个。

You You definitely need some technical ability, and I think that's what people usually think about when they think about strong mathematics.

你能解决这个问题吗？

Like, can you solve this problem?

你能证明这个定理吗？

Like, can you prove this theorem?

但事实上，真正有影响力的研究，往往并不在于解决具体问题或证明定理。

But I I think in practice, the most influential mathematics is not actually about solving problems or or proving a theorem.

它关乎一些更难捉摸的东西，比如某种哲学，甚至带点神秘色彩。

It's about something that's a little bit less tangible, like some kind of philosophy or or mysticism even.

当你试图构建一个理论，或者弄清楚为什么某件事成立时，往往需要发现一种新的结构。

Like, when you're trying to develop a theory or figure out why something is true, often that involves, like, discovering a new structure.

这种结构甚至可能并不十分精确。

That structure might not even be something that is really precise.

你把一种理念传递给其他数学家，帮他们找到理解新数学对象的方式。

Like, you spread a philosophy to a community of other mathematicians and give them a way of understanding new mathematical objects.

我觉得，这根本不是纯粹形式化的东西。

I think you don't really think of that as it's not a purely formal thing.

你在中学、大学或者任何阶段学习数学时，通常不会觉得这是数学的一部分。

You don't necessarily think of that as being mathematics when you're in high school or middle school or college or whatever.

但我觉得，数学里最重要的部分，其实更接近哲学，而不是科学。

But, yeah, the most important parts of mathematics are actually closer to philosophy, I think, than science.

最后要强调的一点是创造力。

Well, the final thing to highlight is creativity.

伟大的想法往往是在数学家之间的对话中产生的。

And where great ideas often arise is really in conversation between mathematicians.

我们经常四处奔波，就是为了面对面地在黑板前讨论数学问题。

I mean, we travel all the time to speak to each other in person at a chalkboard about some mathematical ideas.

在合作中，几个头脑碰撞在一起，突然就冒出一个新的方向，这真的很奇妙。

And it's amazing in a collaboration that a new direction can just originate spontaneously from a bringing together of multiple minds.

我太喜欢这一点了。

I love that.

艾米莉和丹尼尔，谢谢你们今天抽出时间跟我们交流。

Emily and Daniel, thank you for taking the time to talk to us today.

谢谢你们的邀请。

Thanks for having us.

谢谢。

Thank you.

对，这真是一次愉快的交谈。

Yeah, that was fun.

这里是约翰斯·霍普金斯大学的艾米莉·里尔博士，以及多伦多大学的丹尼尔·利特博士。

Doctor Emily Riehl, mathematician at Johns Hopkins University, and Doctor Daniel Litt, mathematician at the University of Toronto.

本集节目由彼得·施密特制作。

This episode was produced by d Peter Schmidt.

如果你没想到一场关于数学的对话会让你如此动情，欢迎留下评价，点击关注或订阅你常用的收听平台。

If you're surprised to find that you got all the feels from a conversation about math, please leave a review and tap, follow, or subscribe wherever you listen.

感谢收听。

Thank you for listening.

我是弗洛拉·利希特曼。

I'm Flora Lichtman.

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